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\title{ HW I for Math 6210\\ due W Sep. }
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\noindent 
1)  Let $\mathfrak M$ be an infinite $\sigma$-algebra. Prove that 
\begin{itemize} 
\item $\mathfrak M$ contains an infinite sequence of disjoint sets. 
\item $\mathrm{card}(\mathfrak M) \geq \mathfrak c$. 
\end{itemize} 

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\noindent 
2) Let $\mathcal E_1\subset P(X_1)$ and $\mathcal E_2\subset P(X_2)$ two elementary families. Let $\mathcal E_1 \times \mathcal E_2$ be the collection of 
all products $E_1\times E_2$ where $e_1\in \mathcal E_1$ and $E_2 \in \mathcal E_2$. Prove that this is an elementary family i
of sets in $X_1 \times X_2$. 

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\noindent 
3) 
An algebra $\mathcal A$ is a $\sigma$-algebra if and only if it is closed under countably increasing unions i.e. if $E_1 \subseteq E_2 \subseteq \ldots \in \mathcal A$ then 
$\cup_{i=1}^{\infty} E_i\in \mathcal A$. Similarly, a finitely additive measure $\mu$  on a $\sigma$-algebra $\mathcal M$ is a measure if and only if for any 
$E_1 \subseteq E_2 \subseteq \ldots \in \mathcal M$, $\lim_{i\to \infty} \mu(E_i) = \mu( \cup_{i=1}^{\infty} E_i)$. 

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\noindent 
4) Suppose $(X,\mathcal M, \mu)$ is a measure space. Let $\mathcal N$ be the collection of all sets $N\in \mathcal M$ of measure 0. Let 
$\overline{\mathcal M}$ be the collection of unions $E\cup F$ where $E\in \mathcal M$ and $F\subseteq N$ for some $N\in \mathcal N$. 
Define $\bar{\mu}(E\cup F) =\mu(E)$. Prove that $(X, \overline{\mathcal  M}, \bar{\mu})$ is a measure space. 


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\noindent 
5) Let $X=(0,1]$. Let $\mathcal A$ be the algebra consisting of disjoint unions of intervals $(a,b]$, with the usual pre-measure 
defined by $\mu_0((a,b])=b-a$. Let $\mu^*$ be the corresponding outer measure. Let $A\subset X$. 
Prove that $\mu^*(A) + \mu^*(A^c)=1$ if and only if for every $\epsilon>0$, there exists $E\in \mathcal A$ such that $\mu^*(A\Delta E) < \epsilon$. 


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\noindent 
6) The setting as in the previous exercise. Prove that $A\subset X$ is measurable if and only if $\mu^*(A) + \mu^*(A^c)=1$. 


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\noindent 
7) Let $\mu$ be a finite Borel measure on $\mathbb R$. Let $F(x)= \mu(-\infty, x]$. Show that $F(x)$ is continuous at $x$ if and only if $\mu(\{x\})=0$. 




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